# proof of cosines law

Let $a$, $b$, $c$ be the sides of a triangle and $\alpha$, $\beta$, $\gamma$ its angles, respectively.  By the projection formula, one may write the equalities

 $\displaystyle\begin{cases}a=b\cos\gamma+c\cos\beta\\ b=c\cos\alpha+a\cos\gamma\\ c=a\cos\beta+b\cos\alpha.\end{cases}$

Multiplying the equalities by $a$, $-b$ and $-c$, respectively, they read

 $\displaystyle\begin{cases}a^{2}\;=\;ab\cos\gamma+ca\cos\beta\\ -b^{2}=-bc\cos\alpha-ab\cos\gamma\\ -c^{2}=-ca\cos\beta-bc\cos\alpha.\end{cases}$

Addition of these yields the sum equation

 $a^{2}-b^{2}-c^{2}=-2bc\cos\alpha,$

i.e.

 $a^{2}\;=\;b^{2}+c^{2}-2bc\cos\alpha,$

which is the cosines law.

Title proof of cosines law ProofOfCosinesLaw 2013-03-22 18:27:13 2013-03-22 18:27:13 pahio (2872) pahio (2872) 4 pahio (2872) Proof msc 51M04 DerivationOfCosinesLaw